ELASTIC CONSTANTS AND LOSSES IN NICKEL 987 2.5 X lfl" 4 , and use of equation (2) with this and other appropriate values indicates that the domain size is / - 0.035 mm, (21) as reported above. A check on this value can be obtained from the frequency, f m , corre- sponding to the maximum of the d vs/ curve. If we use equation (16), Prfm/R- 0.13, (22) with/- 1.5X 10* (Fig. 11), we find f« 0.045 mm, in reasonable agreement. An actual photograph of domains in a single crystal of nickel, taken by 11. J. Williams and reproduced in Fig. 13, shows the presence of domains of various sizes ranging from about 0,01 to 0*2 mm. Any such range in domain sizes will naturally tend to flatten the maximum of the 6 vs/ curve and, on account of the form of the S vs/ function, will push the maximum to a higher frequency than that corresponding to the initial slope, and will give a lower maximum value to the decrement frequency curve. The average domain size derived from our experiments is somewhat larger than that previously obtained in 68 Permalloy* 6 This may be expected, for nickel has a very high magnetostriction and the movement of domain boundaries by stress will he relatively large, possibly so large that the re- gions swept over by the domain walls will correspond to whole domains of the original domain structure, when the stresses are equal to those used in our experiments. The domain size which we have determined is based on this interpretation. APPENTJDt METHOD OK MEASUREMENT— FORMULAE From transmission line theory (see reference of footnote 12) the ratio of outputs, r, define*! in the text and applicable to the circuit of Fig. 9 is given by c«fa «p + r ( =r + ~) sinh AC, (24) where = A + jB = propagation constant ^° = a _s_ •» = characteristic impedance of rod A *f JD Z r — resistive terminating impedance provided by crystals S = area of rod This expression may be expanded into real and imaginary parts and the latter term set to zero in accordance with the condition of phase balance. 088 THE BEU SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 Assuming that [ Z, \ » Z T and that Q - jj > 10, the ratio r which is now a real number determines the attenuation A in accordance with equation (13) of the text. If the crystal resonance frequency/, is slightly different from the balance frequency /, obtained with the rod specimen in place, correction may be made by considering a new terminating resislance Z' T formed by the crystal driver and a small section of the rod sufficient to mike the combined reso- nance equal to/, . A slightly different length. l' a , of rod h then used to compute velocities and attenuation. Also a different mass M„ and ratio r result. The equation applicable provided /,= /, , is Sinh At. - "grfr'-coriiXQ (25) where *-*[»HS(H] (26) ?. In the above a sufficiently accurate value of Q is ordinarily obtained by assuming / e - /- • Further accuracy, if needed, can be obtained by recal- culation, using the corrected value of Q. We are glad to acknowledge the cooperation of Mr. J. G. Walker in grow- ing and processing the single crystals used/ and in preparing also the poly- cryslallinc specimens. References l t H. f, McSkimtn, Jl Acms. Soc. Amtr. t 22, 413 (1050). particularly method K. 2. R. M. Botorlli, W. P. Mason. IL J. McSkimin, J. G. Walker. Phys. Jta, ?S t 1954 (1049). « , _ « 3. Summarized by (a) R. Becker, and \V. Mring, Fcrromaenemmus, Spnngcr, Berlin (1939), and by (») R. M. Bozorih, Fcrrooiagoclism, Van Noslrand, New \ork (1951). 4. W, P. Mason, Phyt. tee. 83, 683 (1951). 5. IL I. Williams and R. M. Boxorlh, Phyt. R*9. f 50, 939 (1941), and reference 3b. 6. H. J. Williams and J. G. Walker, Pkys. to. SJ t 634 (1951)- RUSTIC CONSTANTS AND LOSSES IM >HCKEL 969 7. J, G. Walker. II. J, Williams and It M. Bozorth, Re*. Sti. Instruments, 20, 947 (1949) ami reference 2. 8. W. P. Mason, Pttys. Rcr. S2 t 715, (1951), 9. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, Van Nostrand, New York (1950). amm m tt t _ 10 K, Honda and Y. Shirakawa, Nippon Kinzoku Gakkai-Si (//. Inst. Metals, Japan) J, 217 (19J7), and ScL fop. Res. lnst. t Tohoku Univ. J, 9 (1949). 11. M. Yamamolo. 71. InsL Ittfult, Japan tf t 3-tl (1942) and Pttys. An, 77 t 560 (1950). 12. This method is a modification o! one described in " Elect rnnicchamcal transducer* and Wave Filters," W. P. Mason, D. Van Nostrum! (1942) pages 244-247. For frequencies aliovc 20 Ice, 45* Z-cul ammonium dihydrogen phosphate C q^tlM were used. Below 20 kc the crystals were cemented on square brass rods which were placed at the ends of the nickel rods in place of the piezoelectric crystals alone, 13. W. Doring, Z. Physit, 114, 579 (1939).